On the collision rate of small particles in isotropic turbulence Part 2. Finite inertia case

Numerical experiments have been performed to study the geometric collision rate of heavy particles with finite inertia. The turbulent flow was generated by integrating numerically the full Navier-Stokes equations directly. We found that the collision kernel reached a peak at a particle response time larger than the Kolmogorov time but less than the large-eddy turnover timeΓimplying that both the large-scale and small-scale fluid motion can contributeΓalthough in very different mannersΓto the collision rate. Both numerical results and a stochastic theory show that the collision kernel approaches very slowly to the kinetic theory of Abrahamson (1975) at large τ_p/T_eΓwhere τ_p is the particle response time and T_e is the flow integral time scale. A rapid increase of the collision kernel with the particle response time was observed for small τ_p/τ_kΓwhere τ_k is the flow Kolmogorov time scale. A small inertia of τ_p/τ_k = 0.5 can lead to an order of magnitude increase in the collision kernel relative to the zero-inertia particles. A scaling law for the collision kernel at small τ_p/τ_k was proposed and confirmed numerically by varying the particle sizeΓinertial response timeΓand flow Reynolds number. A leading-order theory for small τ_p/τ_k was developedΓshowing that the enhanced collision is mainly a result of the nonuniform particle concentration due to the interaction of heavy particles with local flow microstructures.